The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria A Global Theory of Algebras of Generalized Functions M. Grosser M. Kunzinger R. Steinbauer J. Vickers Vienna, Preprint ESI 813 (1999) December 27, 1999 Supported by Federal Ministry of Science and Transport, Austria Available via http://www.esi.ac.at

A global theory of algebras of generalized functions M. Grosser, M. Kunzinger, R. Steinbauer Universitat Wien Institut fur Mathematik J. Vickers University of Southampton, Department of Mathematics Abstract. We present a geometric approach to dening an algebra ^G(M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space D 0 (M) of distributions on M. Based on dierential calculus in convenient vector spaces we achieve an intrinsic construction of ^G(M). ^G(M) is a dierential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector elds. Moreover, we construct a canonical linear embedding of D 0 (M) into ^G(M) that renders C 1 (M) a faithful subalgebra of ^G(M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus ^G(M) retains all the distinguishing properties of the local theory in a global context. 2000 Mathematics Subject Classication. Primary 46F30; Secondary 46T30. Key words and phrases. Algebras of generalized functions, Colombeau algebras, distributions on manifolds, calculus on innite dimensional spaces. 1 Introduction Colombeau's theory of algebras of generalized functions ([6], [7], [8], [23]) is a well-established tool for treating nonlinear problems involving singular objects, in particular, for studying nonlinear dierential equations in generalized functions. A Colombeau algebra G() on an open subset of R n is a dierential algebra containing D 0 () as a linear subspace and C 1 () as a faithful subalgebra. The embedding D 0 (),! G() commutes with partial dierentiation and the functor! G() denes a ne sheaf of dierential algebras. In view of L. Schwartz's impossibility result ([26]) these properties of Colombeau algebras are optimal in a very precise sense (cf. [23], [24]). In the so-called \full version of the theory the embedding D 0 (),! G() is canonical, as opposed to the \special or \simplied version (cf. the remark below), where the embedding depends on a particular mollier. The main interest in the theory so far has come from the eld of nonlinear partial dierential equations (cf. e.g. [3], [4], [10], [11], [12], as well as 1

[23] and the literature cited therein), whereas the development of a theory of Colombeau algebras on manifolds proceeded at a much slower pace. [1] presents an approach which basically consists in lifting the sheaf G from R n to a manifold M. A more rened sheaf-theoretic analysis of Colombeau algebras on manifolds is given in [13]. It treats the \special version of the algebra in the sense of [23], p. 109f, whose elements (termed ultrafunctions by the authors) depend on a real regularization parameter. In both approaches, as well as in the construction envisaged in [2], the canonical embedding C 1 (M),! D 0 (M),! G(M) is lost when passing from M = R n to a general manifold (due to the fact that in these versions action of dieomorphisms commutes with embedding only in the sense of association but not with equality in the algebra). Also, although Lie derivatives of elements of G(M) are dened in [13] the operation of taking Lie derivatives does not commute with the embedding (again this property only holds in the sense of association). To remedy the rst of these defects, J. F. Colombeau and A. Meril in [9] (using earlier ideas of [6]) introduced an algebra of generalized functions on whose elements depend smoothly (in the sense of Silva-dierentiability) on ('; x), where ' 2 D(R n ) and x 2. The aim of [9] is to make the embedding of D 0 () (which is done basically by convolution with the ''s: u 2 D 0 () 7! (('; x) 7! hu(y); '(y? x)i)) commute with the action of dieomorphisms. However, an explicit counterexample in [19] demonstrated that the construction given in [9] is in fact not dieomorphism invariant. Also in [19], J. Jelnek gave an improved version of the theory clarifying a number of open questions but still falling short of establishing the existence of an invariant (local) Colombeau algebra. Finally, this aim was achieved in [16] and [17], where a complete construction of dieomorphism invariant Colombeau algebras on open subsets of R n is developed. Several characterization results derived there will be crucial for the presentation in the following sections. Summing up, the current state of the theory is that a dieomorphism invariant local version (and, therefore, a version on manifolds) of Colombeau algebras providing canonical embeddings of smooth functions and distributions is available. Recent applications of Colombeau algebras to questions of general relativity (e.g. [5], [2], [21], [27], for a survey see [28]) have underscored the need for a theory of algebras of generalized functions on manifolds that enjoys two additional features: rst, it should be geometric in the sense that its basic objects should be dened intrinsically on the manifold itself. Second, the object to be constructed should be a dierential algebra with Lie derivatives commuting with the embedding of D 0 (M). In this paper we construct an algebra ^G(M) satisfying both of these requirements. In particular, elements of ^G(M) possess a Lie derivative ^LX 2

with respect to arbitrary smooth vector elds X on M. Moreover, each Lie derivative commutes with the canonical embedding of D 0 (M) into ^G(M), so that in fact all the distinguishing properties of Colombeau algebras on open subsets of R n are retained in the global case. The key concept leading to a global formulation of the theory is that of smoothing kernels: Denition 3.3 (i) below is in a sense the dieomorphism invariant `essence' of the process of regularization via convolution and linear scaling on R n while 3.3 (ii) is the invariant formulation of the interplay between x- and y-dierentiation in the local context. Finally, a number of localization results in section 4 allow one to make full use of the well-developed local theory also in the global context. 2 Notation and terminology, the local theory Throughout this paper, M will denote an oriented paracompact C 1 -manifold of dimension n. An atlas of M will usually be written in the form A = f(u ; ) : 2 Ag. By n c (M) we mean the space of compactly supported (smooth) n-forms on M. Locally, for coordinates y 1 ; : : : ; y n on U, elements of n c ( (U )) will be written as ' d n y := ' dy 1 ^ ^ dy n. The pullback of any! 2 n c ( (U )) under is written as (!). Then R M (' d n y) = R ' (U) dn y for all ' d n y 2 n( c (U )). For U(open) M open we will notationally suppress the embedding of n(u) into c n c (M), and similarly for the inclusion of D(U) (compactly supported smooth functions with support contained in U) into D(M). We generally use the following convention: if B (U ) then ^B :=?1 (B) and if f : (U )! R resp. C then ^f := f. A similar \hat-convention will be applied to the function spaces to be dened in the following section. Moreover, since M is supposed to be oriented we can and shall identify n-forms and densities henceforth. For an open subset of R n, the space of distributions on (i.e., the dual of the (LF)-space D()) will be denoted by D 0 (). For a dieomorphism : ~!, the pullback of any u 2 D 0 () under is dened by h (u); 'i = hu(y); '(?1 (y)) jdet D?1 (y)ji (1) Concerning distributions on manifolds we follow the terminology of [14] and [22], so the space of distributions on M is dened as D 0 (M) = n c (M)0. Observe that in this setting test objects no longer have function character but are n-forms. In the context of Colombeau algebras it is natural to regard smooth functions as regular distributions (which in the non-orientable case enforces the use of test densities; this is in accordance with [18] but has to be distinguished from the setting of [15]). 3

Operations on distributions are dened as (sequentially) continuous extensions of classical operations on smooth functions. In particular, for X 2 X(M) (the space of smooth vector elds on M) and u 2 D 0 (M) the Lie derivative of u with respect to X is given by hl X u;!i =?hu; L X!i. If u 2 D 0 (M), (U ; ) 2 A then the local representation of u on U is the element (?1 ) (u) 2 D 0 ( (U )) dened by h(?1 ) (u); 'i = huj U ; ('dn y)i 8' 2 D( (U )) (2) It should be clear from (1) and (2) that the character of test objects as n- forms is actually already built into the local theory of distributions on R n (i.e., on the right hand side of (1) u acts exactly on the coecient function of (?1 ) (' d n y)). As in [16], [17] dierential calculus in innite dimensional vector spaces will be based on the presentation in [20]. The basic idea is that a map f : E! F between locally convex spaces is smooth if it transports smooth curves in E to smooth curves in F, where the notion of smooth curves is straightforward (via limits of dierence quotients). The dieomorphism invariant theory of Colombeau algebras on open subsets of R n introduced in [16] is based on this notion of dierentiability. Of the two (equivalent) formalisms for describing the dieomorphism invariant local Colombeau theory analyzed in [16], Section 5 only one (the so-called J-formalism which was employed by Jelnek in [19]) lends itself naturally to an intrinsic generalization on manifolds, as the embedding of distributions does not involve a translation (see below). We briey recall the main features of this theory. Let R n open; then dene A 0 () = f' 2 D()j A q (R n ) = f' 2 A 0 (R n )j '()d = 1g '() d = 0; 1 jj q; 2 N n 0g (q 2 N) : The basic space of the dieomorphism invariant local Colombeau algebra is dened to be E() = C 1 (A 0 () ). The algebra itself is constructed as the quotient of the space of moderate modulo the ideal of negligible elements R of the basic space where the respective properties are dened by plugging scaled and transformed \test objects into R and analyzing its asymptotic behavior on these \paths as the scaling parameter tends to 0. Dieomorphism invariance of the whole construction is achieved by dieomorphism invariance of this process termed \testing for moderateness resp. negligibility in [16], Section 9. We shall discuss this matter in some more detail at the end of this section but now proceed by dening the actual \test 4

objects. Set I = (0; 1]. Let C 1(I ; A b 0(R n )) be the space of smooth maps : I! A 0 (R n ) such that for each K (i.e., K a compact subset of ) and any 2 N n 0, the set f@ x (; x) j 2 (0; 1]; x 2 Kg is bounded in D(R n ). For any m 1 we set A 2 m() := f 2 C 1 b A m () := f 2 C1 b (I ; A 0 (R n ))j sup j x2k (I ; A 0(R n ))j sup j x2k (; x)() dj = O( m ) (1 jj m) 8K g (; x)() dj = O( m+1?jj ) (1 jj m) 8K g Elements of A 2 m() are said to have asymptotically vanishing moments of order m (more precisely, in the terminology of [17], elements of A 2 m() are of type [A g ], the abbreviation standing for asymptotic vanishing of moments globally, i.e., on each K ). The spaces A 2 m() and A m() play a crucial role in the characterizations of the algebra ([16], Section 10) and in Section 4 below (cf. also the discussion of dieomorphism invariance at the end of this section). Also for later use we note that A 2 m() A m() A 2 2m?1(). For x 2 R n, 2 I we dene translation resp. scaling operators by T x : D(R n )! D(R n ), T x (') = '(:? x) and S : D(R n )! D(R n ), S ' =?n '( : ). Now the subspaces of moderate resp. negligible elements of E() are dened by E m () = fr 2 E() j 8K 8 2 N n 0 9N 2 N 8 2 C 1 b (I ; A 0(R n )) : sup j@ (R(T x S (; x); x))j = O(?N ) (! 0)g x2k N () = fr 2 E() j 8K 8 2 N n 0 8r 2 N 9m 2 N 8 2 C 1 b (I ; A m (R n )) : sup j@ (R(T x S (; x); x))j = O( r ) (! 0)g x2k By [16], Th. 7.9 and [17], Corollaries 16.8 and 17.6, R 2 E m () is negligible i 8K 8 2 N n 0 8r 2 N 9m 2 N 8 2 A 2 m(); 9C > 0 9 > 0 8 (0 < < ) 8x 2 K: j@ (R(T x S (; x); x))j C r : The dieomorphism invariant Colombeau algebra G() on is the quotient algebra E m ()=N (). Partial derivatives in G() are dened as (D i R)('; x) =?((d 1 R)('; x))(@ i ') + (@ i R)('; x) (3) where d 1 and @ i denote dierentiation with respect to ' and x i, respectively. Note that formula (3) is exactly the result of translating the usual partial 5

dierentiation from the C-formalism to the J-formalism (cf. [16], Section 5). With these operations,! G() becomes a ne sheaf of dierential algebras on R n. For R 2 E m () we denote by cl[r] its equivalence class in G(). The map : D 0 ()! G() u! cl[('; x)! hu; 'i] provides a linear embedding of D 0 () into G() whose restriction to C 1 () coincides with the embedding : C 1 ()! G() f! cl[('; x)! f(x)] so renders C 1 () a faithful subalgebra and D 0 () a linear subspace of G(). Moreover, commutes with partial derivatives due to the specic form of (3). The pullback of R 2 E() under a dieomorphism : ~! is dened as (^R)( ~'; ~x) = R(( ~'; ~x)) where ( ~'; ~x) = (( ~'?1 ) j det D?1 j; (~x)). Pullback under dieomorphisms then commutes with the embedding of D 0 () into G(). Finally we return to the issue of dieomorphism invariance. Since the action of a dieomorphism on R 2 E() is dened in a functorial way dieomorphisminvariance of the denition of the moderate resp. negligible elements of E() indeed is invariant if the class of scaled and transformed \test objects is; more precisely if (; x) = S?1 T?x pr 1 (T?1 x S (; ~?1 x)();?1 x) = (; ~?1 ( + x)??1 x?1 x) j det D?1 ( + x)j (4) is a valid \test object if ~ (; ~x) was. However, if ~ 2 C 1 b (I ; A q(r n )) in general will neither be dened on all of I nor will its moments be vanishing up to order q. To remedy these defects a rather delicate analysis of the testing procedure is required: Denote by C 1 b;w (I ; A 0(R n )) the space of all : D! A 0 (R n ) where D is some subset (depending on ') of (0; 1] and for D; ' the following holds: For each L there exists 0 and a subset U of D which is open in (0; 1] such that (0; 0 ] L U( D) and is smooth on U (5) f@ (; x) j 0 < 0 ; x 2 Lg is bounded in D(R n ) 8 2 N n 0 (6) 6

(the subscript w signies the weaker requirements on the domain of denition of ). Then by [16], Th. 10.5 R 2 E() is moderate i 8K 8 2 N n 0 9N 2 N 8 2 C 1 b;w (I ; A 0(R n )); : D! A 0 (R n ))9C > 0 9 > 0 8 (0 < < ) 8x 2 K: (; x) 2 D and j@ (R(T x S (; x); x))j C?N Moreover, by [16], Th. 7.14, (4) is an element of C 1 b;w (I ~ ; A 0 (R n )) for every 2 C 1 b (I ; A 0(R n )). The subspace of C 1 b;w (I ; A 0(R n )) consisting of those whose moments up to order m vanish asymptotically on each compact subset of will be written as A 2 m;w() (also, A m;w() is dened analogously). By the proof of [16], Cor. 10.7 R 2 E m () is negligible i 8K 8 2 N n 0 8r 2 N 9m 2 N 8 2 A 2 m;w(); : D! A 0 (R n )) 9C > 0 9 > 0 8 (0 < < ) 8x 2 K: (; x) 2 D and j@ (R(T x S (; x); x))j C r and if 2 A 2 m;w() then (4) denes an element of A 2 [ m+1 2 ];w( ~). These facts directly imply dieomorphism invariance of local Colombeau algebras ([16], Thms. 7.15 and 7.16). The dieomorphism invariance of the scaled, transformed \test objects demonstrated above suggests to choose an analogue of these as the main \test objects on the manifold, where no natural scaling resp. translation operator is available. It is precisely this notion which is captured in the denition of the smoothing kernels below. 3 Smoothing kernels and basic function spaces In this section we introduce the basic denitions and operations needed for an intrinsic denition of algebras of generalized functions on manifolds. 3.1 Denition ^A 0 (M) := f! 2 n c (M) :! = 1g The basic space for the forthcoming denition of the Colombeau algebra on M is dened as follows: 3.2 Denition ^E(M) = C 1 ( ^A 0 (M) M) 7

Then (?1 )( ^A 0 ( (U )) (U )) ^A 0 (M) M and locally we have ^E( (U )) = E( (U )), the isomorphism being eected by ^E( (U )) 3 R 7! [('; x) 7! R(' d n y; x)]. In what follows, we will therefore use ^E( (U )) and E( (U )) interchangeably. Clearly the map?1 : n( c (U )) (U )! n (U c ) U,! n (M) M c is smooth. Therefore, for any R 2 ^E(M) its local representation (?1 )^R := R (?1 ) is an element of ^E( (U )). More generally, if : M 1! M 2 is a dieomorphism and R 2 ^E(M 2 ) then its pullback ^(R) 2 ^E(M 1 ) under is dened as R where (!; p) = ((?1 )!; (p)) and clearly ( 1 2 )^= ^ 2 ^ 1 for dieomorphisms 1, 2. Let f : M M! Vn T M be smooth such that for each (p; q) 2 M M, f(p; q) belongs to the ber over q or, equivalently, that for each xed p 2 M, f p := (q 7! f(p; q)) represents a member of n (M). Obviously, p 7! f p 2 C 1 (M; n (M)) in this case. Given a smooth vector eld X on M, we dene two notions of Lie derivatives of f with respect to X which, essentially, arise as Lie derivatives of q 7! f(p; q) resp. p 7! f(p; q): On the one hand, viewing n (M) together with the topology of pointwise (i.e., berwise) convergence as a locally convex space we dene (L 0 Xf)(p; q) := L X (p 7! f(p; q)) = d dt on the other hand, we set (L X f)(p; q) := L X (q 7! f(p; q)) = d dt 0 0 (f(fl X t )(p); q); (7) ((Fl X t ) f p )(q) (8) where the latter symbol L X denotes the usual Lie derivative on the bundle n (M). Now we are ready to introduce the space of smoothing kernels which will serve as an analogue for the (unbounded) sequences (' ) used in [7]. 3.3 Denition 2 C 1 (I M; ^A 0 (M)) C 1 (I M M; n T M) is called a smoothing kernel if it satises the following conditions (i) 8K M 9 0, C > 0 8p 2 K 8 0 : supp (; p) B C (p) 8

(ii) 8K M 8k; l 2 N 0 8X 1 ; : : : ; X k ; Y 1 ; : : : ; Y l 2 X(M) sup kl Y1 : : : L Y l (L0 X 1 + L X1 ) : : : (L 0 + L Xk Xk )(; p)(q)k = O(?(n+l) ) p2k q2m The space of smoothing kernels on M is denoted by ~ A 0 (M). In (i) the radius of the ball B C (p) has to be measured with respect to the Riemannian distance induced by a Riemannian metric h on M. By Lemma 3.4 below, (i) then holds in fact for the distance induced by any Riemannian metric h 0 with a new set of constants 0 (h 0 ) and C(h 0 ). Similarly in (ii), k : k denotes the norm induced on the bers of n c (M) by any Riemannian metric on M (i.e. convergence with respect to k : k amounts to convergence of all components in every local chart). Thus both (i) and (ii) are independent of the Riemannian metric chosen on M, hence intrinsic. 3.4 Lemma Let M be a smooth paracompact manifold and let h 1, h 2 be Riemannian metrics on M. Then for all K M there exist 0 (K) and C > 0 such that 8p 2 K 8 0 : B (2) (p) B (1) C (p) where B (i) (p) = fq 2 M : d i (p; q) < g and d i denotes Riemannian distance with respect to h i. Proof. We rst show that 8K M 9C 0 such that h 1 (p)(v; v) Ch 2 (p)(v; v) 8p 2 K 8v 2 T p M: (9) Without loss of generality we may assume K U for some chart ( ; U ). Denoting by h i (i = 1; 2) the local representations of h i in this chart it follows that f : (x; v)! h 1 (x)(v; v) h 2 (x)(v; v) is continuous (even smooth) on (U ) R n n f0g. Thus sup f(x; v) = x2 (K) v2rn nf0g sup x2 (K) v2@b 1 eucl (0) f(x; v) < 1 with B eucl 1 (0) the Euclidian ball of radius 1 around 0. Next we choose a geodesically convex (with respect to h 2 ) relatively compact neighborhood U p of p 2 M (cf. e.g. [25], Prop. 5.7). Moreover, let C be the 9

square root of the constant in (9) with K = U p. Given any q; q 0 2 U p let be the unique h 2 -geodesic in U p connecting q and q 0. Then d 1 (q; q 0 ) L 1 () CL 2 () = Cd 2 (q; q 0 ) where L i denotes the length of with respect to h i. Now for each p 2 K there exist U p and C p as above and we choose p such that B (2) p (p) U p. Then there exist some p 1 ; : : : ; p m in K with K S m i=1 B(2) p i (p i ) =: U and we set 0 := min(dist 2 (K; @U); p 1 ; : : : ; pm ) and 2 2 2 C := max 1im C p i. Let p 2 K, 0, q 2 B (2) (p). There exists some i with d 2 (p; p i ) p i 2 and by construction d 2 (p; q) p i, so p; q 2 B(2) 2 p i (p i ) U p i. Hence d 1 (p; q) C p id 2 (p; q) and, nally, q 2 B (1) (p). 2 C The next step is to introduce the following grading on the space of smoothing kernels. 3.5 Denition For each m 2 N we denote by ~ A m (M) the set of all 2 ~A 0 (M) such that 8f 2 C 1 (M) and 8K M sup jf(p)? p2k M f(q)(; p)(q)j = O( m+1 ) 3.6 Remark. 3.5 is modelled with a view to reproducing the main technical ingredient for proving j C 1 = (i.e. the fact that the embedding of distributions into G coincides with the \identical embedding on C 1, cf. e.g., [16], Th. 7.4 (iii)) in the local theory. Essentially, the argument in the R local case consists in (substitution of y 0 = y?x and) Taylor expansion of (f(x)? f(y))?n '( y?x )dy (' 2 A 0) yielding appropriate powers of as to establish this term to be negligible. In fact, this argument is at the very heart of Colombeau's construction and may be viewed as the main technical motivation for the concrete form of the sets A q and, a fortiori, of the ideal N. In 3.5 we turn the tables and dene the sets ~A m (M) by the analogous estimate. Moreover, as we shall see shortly (cf. 4.2), locally there is a one-toone correspondence between elements of Am ~ (M) and elements of A m, so the two approaches are in fact equivalent (although only one of them, namely the one occurring in 3.5 admits an intrinsic formulation on M). We are now in a position to prove the nontriviality of the space of smoothing kernels as well as of the spaces ~ A m (M): 3.7 Lemma (i) ~ A0 (M) 6= ;. 10

(ii) ~A m (M) 6= ; (m 2 N). Proof. (i) Let (U ; ) 2A be an oriented atlas of M such that each U is relatively compact. Let f^ j 2 Ag be a subordinate partition of unity and pick ' 2 A 0 (R n ). For each, choose ^ 1 2 D(U ) such that ^ 1 1 in an open neighborhood W U of supp ^. Let := ^?1, 1 := ^ 1?1. Now set 0 (; x)(y) :=?n '( y? x ) : There exists some 0 = 0(supp ) in (0; 1] such that for each x 2 supp( ) and each 0 < we have supp 0 0 (; x) (W ). Choose : R! [0; 1] smooth, 1 on (0; 0 ] and 3 0 on ( 0 ; 1]. Finally, let! 2 ^A 2 0 (M). Then we dene our prospective smoothing kernel by (; p)(q) := X ^ (p) ()? 0 (; (p))( : ) 1 ( : )d n y + (1? ())! (q)(10) R By construction, is smooth and (; p) = 1 for all 2 (0; 1] and all p 2 M. For any given K M, set K := min 0 where ranges over 3 the (nitely many) with K \ U 6= 0; then the terms in (10) containing! vanish for p 2 K and K. In order to prove 3.3 (i), let K M. Since for p 2 K and K, supp (; p) is contained in a nite union of supports of q 7! [? 0 (; (p))( : ) 1 ( : )dn y ](q) it suces to consider only one term of the latter form. Clearly, there exist C and (0 <) 0 ( K ) such that supp 0 (; x) B eucl C (x) for x 2 supp and all 0. Extending the pullback under of a suitable cut-o of the Euclidian metric on (U ) to a Riemannian metric on M, the result follows from 3.4. S m Concerning 3.3 (ii), since K = K i=1 i, K i U i, it suces to estimate on each K i (0; i ] for some i > 0. Thus we may assume that K U 0 for some xed 0. Let L be a compact neighborhood of K in U 0 and let L in what follows. Each of the terms in (10) for which supp ^ does not intersect K vanishes in some open neighborhood of K and can therefore be neglected. Each of the nitely many remaining terms vanishes outside supp ^. Since K (K n supp ^ ) [ (K \ W ) it is sucient to let p range over K \ W in the estimate. Now for p 2 L, 1 can be omitted from the 11

corresponding term in (10). Thus, using local coordinates of (U 0 ; 0 ) we have to estimate a nite number of Lie derivatives of terms of the form ~y? ~x (?1 ) ~y 7!?n ' d n ~y?1 (y)? ~x = y 7!?n ' det D?1 (y) d n y ; (11) each for ~x 2 (K \ W ), respectively, where = 0?1 and (~z) = z. Note that for ~x in a compact subset of (U 0 \ U ) and suciently small (say, 1 ( K )) the support of ~y? ~x ~y 7!?n ' d n ~y is contained in (U 0 \ U ), rendering (11) well-dened. Setting?1 (x + y)??1 (x) (; x)(y) := ' det D?1 (x + y) ; (12) the coecient of the right hand side of (11) can be written as T x S (; x). By [16], Th. 7.14 and the remark following it, for each compact set L there exists 2 ( 1 ) such that f@ x(; x)( : ) j 0 < 2 ; x 2 Lg is (dened and) bounded in D for each 2 N n 0. Of course (12) corresponds to (2) in [9] resp. to (42) in [19], the only dierence being that since we use an oriented atlas and our test objects are forms the determinants are automatically positive. Since L X ( d n y) = (L X ) d n y + L X (d n y), in order to verify 3.3 (ii) we have to estimate terms of the form L Y1 : : : L Yl 0 (L 0 X 1 + L X1 ) : : : (L 0 X k 0 + L Xk 0)T x S (; x)(y) (13) where X i = nx a i ri @ ri Y j = nx ri=1 sj =1 b j sj @ sj are local representations in i(u i) of vector elds on M and 0 k 0 k, 0 l 0 l. Explicitly, (13) is given by l Y 0 j=1 0 X @ n sj=1 b j sj (y) @ 1 Y A k 0 @y s j i=1 nx ri=1! a i (x) @ ri @x + r i ai (y) @ T ri @y r x S (; x)(y): i 12

@ Note that @y s j T x S =?1 @ T x S @y s j and a i (x) @ ri @x + r i ai (y) @ T ri @y r x S = i T x S a i (x) @ ri @x + 1 r i (ai (x + y)? ri ai (x)) @ : ri @y r i Each of the maps (; x; y)! 8 < : a i (x + y)? ri ai (x) ri for 6= 0; (14) Da i (x)y ri for = 0 is smooth (hence uniformly bounded) on each relatively compact subset of its domain of denition. Due to the boundedness of f@ x(; x)(:) j 0 < 2 ; x 2 0 (K \ W )g only values of y from a bounded region of R n are relevant in (14). Thus one further restriction of the range of establishes the claim. (ii) Choose the 0 as in (i), yet this time additionally requiring ' 2 A m(r n ). To estimate sup p2k j R M (; p)(q) ^f(q)? ^f(p)j for some K M and ^f 2 C 1 (M) again only nitely many terms of (10) have to be taken into account; also, for small, the terms involving! vanish and 1 can be neglected. Then sup j p2k\supp ^ M = sup j x2 (K\supp ^) ( 0 (; (p))(:) d n y)(q) ^f(q)? ^f(p)j = (U) 1 '(y? x )f(y) d n y? f(x)j = O( m+1 ) n 2 Next, we introduce the appropriate notion of Lie derivative for elements of ^E(M). 3.8 Denition For any R 2 ^E(M) and any X 2 X(M) we set (^L X R)(!; p) :=?d 1 R(!; p)(l X!) + L X (R(!; : ))j p (15) Here, d 1 R(!; x) denotes the derivative of!! R(!; x) in the sense of [16], section 4. In order to obtain a structural description of this Lie derivative (which, at the same time, entails ^L X R 2 ^E(M) for R 2 ^E(M)), for any X 2 X(M) dene X A 2 X( ^A 0 (M)) by X A : ^A 0 (M)! ^A 00 (M) X A (!) =?L X! 13

(where ^A 00 (M) is the linear subspace of n c (M) parallel to ^A 0 (M), i.e. ^A 00 (M) = f! 2 n c (M)j R! = 0g). Then X A R( : ; p)(!) = d 1 R(!; p)(?l X!), so that (^L X R)(!; p) = (L (X A;X)R)(!; p) = (L (?L X : ;X)R)(!; p) i.e. ^LX is the Lie derivative of R with respect to the smooth vector eld (X A ; X) on ^A 0 (M) M. Thus, indeed ^LX R 2 ^E(M). 3.9 Remark. (Local description of ^L X ) Let X 2 X(M) and set X = (?1 ) (Xj U ). Since the derivative of the (linear and continuous) map : n( c (U ))! n(u c ) in any point equals the map itself, on U we obtain (writing R in place of Rj U ): ((?1 )^(^LX R))(' d n y; x) = (^LX R)( (' dn y);?1 (x)) = L X (R( (' d n y); : ))(?1 (x))? (d 1 R)( (' d n y);?1 (x))(l X ( (' d n y)) ) {z } (L X (' dn y)) = [L X ((?1 )^R))](' d n y; x)? [d 1 ((?1 )^R)(' d n y; x)](l X (' d n y)) In particular, if X = @ y i (1 i n) then this exactly reproduces the local algebra derivative with respect to y i given in (3). Finally we are in a position to dene the subspaces of moderate and negligible elements of ^E(M). 3.10 Denition R 2 ^E(M) is moderate if 8K M 8k 2 N 0 9N 2 N 8 X 1 ; : : : ; X k 2 X(M) 8 2 ~ A 0 (M) sup jl X1 : : : L (R((; p); p))j = X k O(?N ) (16) p2k The subset of moderate elements of ^E(M) is denoted by ^E m (M). 3.11 Denition R 2 ^E(M) is called negligible if it satises 8K M 8k; l 2 N 0 9m 2 N 8 X 1 ; : : : ; X k 2 X(M) 8 2 ~ A m (M) sup jl X1 : : : L (R((; p); p))j = X k O(l ) (17) p2k The set of negligible elements of ^E(M) will be denoted by ^N (M). 14

4 Construction of the algebra, localization The following result is immediate from the denitions: 4.1 Theorem (i) ^E m (M) is a subalgebra of ^E(M). (ii) ^N (M) is an ideal in ^E m (M). For the further development of the theory it is essential to achieve descriptions of both moderateness and negligibility that relate these concepts to their local analogues [19, 16], thereby making available the host of local results already established for open subsets of R n also in the global context. As a rst step, we are going to examine localization properties of smoothing kernels: 4.2 Lemma Denote by (U ; ) a chart in M. (A) Transforming smoothing kernels to local test objects. (i) Let be a smoothing kernel. Then the map dened by (; x)(y)d n y := n ((?1 ) (;?1 x))(y + x) (x 2 (U ); y 2 R n ) (18) is an element of C 1 b;w (I (U ); A 0 (R n )). (ii) If, in addition, 2 ~A m (M) for some m 2 N then 2 A m;w ( (U )), i.e., (; x)(y)y dy = O( m+1?jj ) (1 jj m) (19) uniformly on compact sets. In particular, if 2 ~ A 2m?1 (M) then 2 A 2 m;w( (U )). (B) Transporting local test objects onto the manifold. (i) Let 2 C 1 b (I ; A 0(R n )) and 1 2 ~A 0 (M). Let K (U ), ; 1 2 D( (U )) with 1 on an open neighborhood of K and 1 1 on an open neighborhood of supp. (; p) := (1? ^(p)()) 1 (; p) 1 +^(p)() (; p)( y? p n 2 ) 1 (y)d n y (20) is a smoothing kernel (the smooth cut-o function is dened in the proof). 15

(ii) If, in addition, 2 A m ( (U )) and 1 2 ~A m (M) then 2 ~A m (M). In particular, if 2 A 2 m( (U )) and 1 2 ~ A m (M) then 2 ~ A m (M). Proof. (A) (i) Let ^D1 := f(; p) 2 I U : supp (; p) U g and set D = int((id )( ^D 1 )). Then evidently is smooth on D. Furthermore (5) is obvious from (i) in 3.3. Concerning (6), set Then we have 0 (; x)d n y := (?1 ) ((;?1 (x))) ((; x) 2 D 1 ) @ y @ x = @ y @ xs?1 T?1 x 0 = jj S?1 T?1 x @ y (@ x + @ y ) 0 : (21) Let now K (U ), x 2 K and (K). By Denition 3.3 (i) and Lemma 3.4 there exists a constant C such that diam(supp 0 (; x)) C for all S x 2S K. But then diam(supp (; x)) C for all such x. Hence supp x2k @ x(; x) is bounded uniformly in. Moreover, observing that L (L @y @x + L @y ) ( 0 (; x)(y)d n y) = (L (L @y @x + L @y ) 0 (; x)(y))d n y, from Denition 3.3 (ii) we obtain @ y (@ x +@ y ) 0 = O(?(jj+n) ) which together with equation (21) gives the desired boundedness property. (ii) Now suppose that 2 ~A m (M) and for each jj m choose f 2 D(U ) D(M) such that f?1 (x) = x in a neighborhood of K (U ). Then by assumption we have sup j x2k M f (q)(;?1 (x))(q)? f (?1 (x))j = O(m+1 ) For suciently small this implies sup j x2k?n (; x)( y? x R n )y dy? x j = O( m+1 ) or, upon substituting z =?1 (y? x) and expanding: X sup x2k jj x? (; x)(z)z dz R = O(m+1 ) n 0< From this, we prove (19) by induction with respect to jj: for = e k (the k-th unit vector) we obtain j R R n (; x)(z)z k dzj = O( m ). Supposing now that (19) has already been proved for jj? 1, the result for 16

jj follows from sup x2k j X 0<< = O( m+1 ) jj x? (; x)(z)z dz + jj (; x)(z)z R R dzj n n {z } O( m+1?jj ) (B) (i) First note that by the boundedness assumption all supports of (I supp()) are in a xed compact subset of R n. By [16], Lemma 6.2 we conclude that? there exists > 0 such that for all and x 2 1 supp supp n (; x)( :?x) f 1 1g. Now we are in the position to dene as follows. Let 2 C 1 (R), 0 1 and R 1 on (?1; =3) and 0 on (=2; 1). This actually implies (; p) = 1 8p;. Smoothness again is evident while condition 3.3 (i) is an easy consequence of [16], Lemma 6.2. 3.3 (ii) then follows exactly as (13) is proved in Lemma 3.7. (ii) Let ^f 2 C 1 (M) and p 2 ^K U ; then j M (; p)(q) ^f(q)? ^f(p)j j1? ^(p)()j j j^(p)()j j M M 1 (; p)(q) ^f(q)? ^f(p)j + (?n (; (p))( y? (p) ) 1 (y)d n y)(q) ^f(q)? ^f(p)j The rst summand is of order m+1 by assumption and the second one (apart from ^ and ) for suciently small and setting f = ^f?1 equals R X @ f(x) (; x)(y)f(x + y)d n y? f(x) = jj (; x)(y)y dy n! so the claim follows. 0<jjm {z } O( m+1?jj ) 2 Restriction of an element R of E(M) to an open subset U of M is dened by Rj U := Rj ^A0 (U )U. 4.3 Theorem (Localization of moderateness) Let R 2 ^E(M). Then R 2 ^E m (M), (?1 )^(Rj U ) 2 E m ( (U )) 8: 17

Proof. ()) For R 2 ^E m (M) and for (U ; ) some chart in M let R 0 := (?1 )^(Rj U ). Let K (U ) =: V, 2 N n 0 and let 2 C 1 b (I (U ); A 0 (R n )). We have to show that sup j@ (R 0 (T x S (; x); x))j = O(?N ) (22) x2k for some N 2 N. To this end we x some 1 2 ~ A 0 (M) and dene 2 ~ A 0 (M) by (20). Let @ = @ i1 : : : @ i jj and for 1 i j n choose X i j 2 X(M) such that the local expression of X i j coincides with @ i j on a neighborhood of K. Then for suciently small (22) equals sup jl X i 1 : : : L X i jj R((; p); p)j ; p2 ^K so we are done. (() Let X 1 ; : : : ; X k 2 X(M) and (without loss of generality) ^K U for some chart (U ; ). Let 2 ~A 0 (M) and dene by (18). Since : D! A 0 (R n ) belongs to C 1 b;w(i (U ); A 0 (R n )) it follows from [16], Th. 10.5 that given 2 N n 0 there exists some N 2 N and some 0 > 0 such that for x 2 K and 0 we have (; x) 2 D and j@ (?1 )^(Rj U )(T x S (; x); x))j = O(?N ). Inserting this into the local representation of (16) immediately gives the result. 2 4.4 Theorem (Localization of negligibility) Let R 2 ^E m (M). Then R 2 ^N (M), (?1 )^(Rj U ) 2 N ( (U )) 8: Proof. ()) Let R 2 ^E m (M), (U ; ) some chart in M and set R 0 := (?1 )^(Rj U ). Let K (U ) =: V, l 2 N and 2 N n 0. Set k = jj and choose m 2 N 0 such that sup jl X1 : : : L (R((; x); x))j = X p2 ^K k O(l ) (23) for all X 1 ; : : : ; X k 2 X(M) and all 2 ~A m (M). Now let 2 A 2 m(v ) and construct from according to (20). By 4.2, 2 ~A m (M), so (with X i j as in the proof of 4.3) sup x2k j@ (R 0 (T x S (; x); x))j = sup jl X i 1 : : : L X i jj (R((; x); x))j = O( l ) p2 ^K Thus the claim follows from the characterization of negligibility following the denition of N () (section 2). 18

(() Let k 2 N 0, l 2 N, X 1 ; : : : ; X k 2 X(M) and ^K U. By the discussion at the end of section 2 there exists m 0 such that (with R 0 = (?1 )^Rj U ) sup j@ (R 0 (T x S (; x); x))j = O( l ) (24) x2k for all jj k and all 2 A 2 m 0 (V ;w ). Now set m = 2m 0? 1 and let 2 ~A m (M). Then dened by (18) is in A 2 m 0 (V ;w ) by 4.2 (A) (ii). Hence inserting local representations of X 1 ; : : : ; X k, (24) immediately implies the validity of (23), thereby nishing the proof. 2 It was shown in [17], sec. 13 that for all variants of (local) Colombeau algebras membership of any element R of E m to the ideal N can be tested on the function R itself, without taking into account any derivatives of R. As a rst important consequence of the above localization results we note that this rather surprising simplication also holds true for the global theory: 4.5 Corollary Let R 2 ^E m (M). Then R 2 ^N (M), (17) holds for k = 0: Proof. This follows directly from Th. 4.4 by taking into account [16], Th. 7.13 and [17], Th. 13.1. 2 Moreover, stability of ^E m (M) and ^N (M) under Lie derivatives also follows from the local description: 4.6 Theorem Let X 2 X(M). Then (i) ^L X ^Em (M) ^E m (M). (ii) ^L X ^N (M) ^N (M). Proof. Let R 2 ^E m (M), X 2 X(M). By Th. 4.3 for any chart (U ; ) we have (?1 )^(Rj U ) 2 E m ( (U )). Thus by [16], Th. 7.10 also L X (?1 )^ (Rj U ) = (?1 )^(^L X Rj U ) 2 E m ( (U )) (where X denotes the local representation of X), which, again by Th. 4.3 gives the result. The claim for ^N (M) follows analogously from [16], Th. 7.11. 2 Finally we are in a position to dene our main object of interest: 4.7 Denition ^G(M) := ^E m (M)= ^N (M) is called the Colombeau algebra on M. 19

By construction, every ^LX induces a Lie derivative (again denoted by ^LX ) on ^G(M), so ^G(M) becomes a dierential algebra. If R 2 ^E(M), its class in ^G(M) will be denoted by cl[r]. 4.8 Theorem ^G(M) is a ne sheaf of dierential algebras on M. Proof. This is a straightforward consequence of [16], Th. 8.1. 2 5 Embedding of distributions and smooth functions In this section we show that in the global context ^G(M) displays the same set of (optimal) embedding properties as the local versions do on open sets of R n. Most importantly, we shall see that taking Lie derivatives with respect to arbitrary smooth vector elds commutes with the embedding. To begin with, let u 2 D 0 (M). The natural candidate for the image of u in ^G(M) is R u (!; x) = hu;!i. We rst show that R u 2 ^E m (M) using Th. 4.3. Let! 2 D( (U )); then ((?1 )^(R u j U ))(!; x) = (R u j U )( (! d n y);?1 (x)) = hu; (! dn y)i = h(?1 ) (uj U );!i Since (?1 ) (uj U ) 2 D 0 ( (U )) it follows from the local theory that indeed (?1)^(R u j U ) 2 E m ( (U )). Suppose now that R u 2 ^N (M). By the same reasoning as above (!; x)! h(?1 ) (uj U );!i 2 N ( (U )) for each. Thus again by the respective local result (?1 ) (uj U ) = 0 for each, i.e. u = 0. Therefore : D 0 (M)! ^G(M) (u) = cl[(!; x)! hu;!i] is a linear embedding. What is more, as a direct consequence of (15) (noting that distributions are linear and continuous, hence equal to their dierential in any point) we obtain (L X u)(!; p) = ((!; p)!?hu; L X!i) =?d 1 R u (!; p)(l X!) + L X (R u (!; : ))j {z } p = (^L X R u )(!; p) = ^L X ((u))(!; p) =0 i.e., commutes with arbitrary Lie derivatives. 20

The natural operation for embedding smooth functions into ^G(M) is given by : C 1 (M)! ^G(M) (f) = cl[(!; x)! f(x)] Obviously, is an injective algebra homomorphism that commutes with Lie derivatives by (15). Moreover, coincides with on C 1 (M). Making use of Th. 4.4 this again follows directly from the local result. Summing up, we have 5.1 Theorem : D 0 (M)! ^G(M), is a linear embedding that commutes with Lie derivatives and coincides with : C 1 (M)! ^G(M) on C 1 (M). Thus renders D 0 (M) a linear subspace and C 1 (M) a faithful subalgebra of ^G(M). The following commutative diagram illustrates the compatibility properties of Lie derivatives with respect to embeddings established in this section: L X C 1 - C 1 J J JJ^ L X D 0 - D 0 J J? JJ^? ^L ^G X - ^G 6 Association The concept of association or coupled calculus is one of the distinguishing features of local Colombeau algebras. It introduces a (linear) equivalence relation on the algebra identifying those elements which are \equal in the sense of distributions, thereby allowing to identify \distributional shadows of certain elements of the algebra. This construction amounts to determining the macroscopic aspect of the regularization procedure encoded in them. Phrased more technically, a linear quotient of E m resp. G is formed containing 21

D 0 as a subspace. Especially in physical modelling this notion provides a useful tool for analyzing nonlinear problems involving singularities (cf. e.g. [5], [21], [27], [28]). In what follows we extend the notion of association to ^G(M). 6.1 Denition An element [R] of ^G(M) is called associated to 0 ([R] 0) if for some (hence every) representative R of [R] we have: 8! 2 n(m) c 9m > 0 with lim R((; p); p)!(p) = 0 8 2 A ~ m (M) (25)!0 M Two elements [R], [S] of ^G(M) are called associated ([R] [S]) if [R?S] 0. We say that [R] 2 ^G(M) admits u 2 D 0 (M) as an associated distribution if [R] (u), i.e. if 8! 2 n c (M) 9m > 0 with lim!0 M R((; p); p)!(p) = hu;!i 8 2 ~A m (M) (26) Finally, by the same methods as in the local theory we obtain consistency in the sense of association of classical multiplication operations with multiplication in the algebra: 6.2 Proposition (i) If f 2 C 1 (M) and u 2 D 0 (M) then (ii) If f; g 2 C(M) then (f)(u) (fu) (27) (f)(g) (fg) (28) References [1] J. Aragona, H. A. Biagioni, Intrinsic denition of the Colombeau algebra of generalized functions, Analysis Mathematica 17 (1991), 75-132. [2] H. Balasin, Geodesics for impulsive gravitational waves and the multiplication of distributions, Class. Quantum Grav. 14 (1997), 455{462. [3] H. A. Biagioni, M. Oberguggenberger, Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations, SIAM J. Math. Anal. 23 (1992), 923{940. 22

[4] H. A. Biagioni, M. Oberguggenberger, Generalized solutions to Burgers' equation, J. Dierential Equations 97 (1992), 263{287. [5] C. J. S. Clarke, J. A. Vickers, J. Wilson, Generalised functions and distributional curvature of cosmic strings, Class. Quantum Grav. 13 (1996), 2485{2498. [6] J. F. Colombeau, New Generalized Functions and Multiplication of Distributions, North Holland, Amsterdam 1984. [7] J. F. Colombeau, Elementary Introduction to New Generalized Functions, North Holland, Amsterdam 1985. [8] J. F. Colombeau, Multiplication of Distributions, Bull. Am. Math. Soc., New Ser. 23, No. 2 (1990), 251-268. [9] J. F. Colombeau, A. Meril, Generalized Functions and Multiplication of Distributions on C 1 Manifolds, J. Math. Analysis Appl. 186 (1994), 357{364. [10] J. F. Colombeau, A. Heibig, M. Oberguggenberger, Le probleme de Cauchy dans un espace de fonctions generalisees I, C. R. Acad. Sci. Paris Ser. I Math. 317 (1993), 851{855. [11] J. F. Colombeau, A. Heibig, M. Oberguggenberger, Le probleme de Cauchy dans un espace de fonctions generalisees II, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), 1179{1183. [12] J. F. Colombeau, M. Oberguggenberger, On a hyperbolic system with a compatible quadratic term: Generalized solutions, delta waves, and multiplication of distributions, Comm. Partial Dierential Equations 15 (1990), 905{938. [13] M. Damsma, J. W. de Roever, Colombeau algebras on a C 1 -manifold, Indag. Mathem., N.S. (3) (1991), 341-358. [14] G. de Rham, Dierentiable Manifolds, Springer, Berlin 1984. [15] J. Dieudonne, Elements d'analyse, Vol 3, Gauthier-Villars, Paris, 1974. [16] E. Farkas, M. Grosser, M. Kunzinger, R. Steinbauer, On the Foundations of Nonlinear Generalized Functions I, Preprint, 1999. [17] M. Grosser, On the Foundations of Nonlinear Generalized Functions II, Preprint, 1999. 23

[18] L. Hormander, The Analysis of Linear Partial Dierential Operators I, Grundlehren der mathematischen Wissenschaften 256, Springer, Berlin, 1990. [19] J. Jelnek, An intrinsic denition of the Colombeau generalized functions, Comment. Math. Univ. Carolinae 40, 1 (1999), 71{95. [20] A. Kriegl, P. W. Michor, The Convenient Setting of Global Analysis, AMS Math. Surv. and Monographs 53, 1997. [21] M. Kunzinger, R. Steinbauer, A rigorous solution concept for geodesic equations in impulsive gravitational waves, J. Math. Phys. Vol. 40, No. 3 (1999), 1479{1489. [22] J. E. Marsden, Generalized Hamiltonian Mechanics, Arch. Rat. Mech. Anal. 28 (1968), 323{361. [23] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Dierential Equations, Pitman Research Notes in Mathematics 259, Longman 1992. [24] M. Oberguggenberger, Nonlinear theories of generalized functions, in: S. Albeverio, W.A.J.Luxemburg, M.P.H.Wol (Eds.), Advances in Analysis, Probability, and Mathematical Physics-Contributions from Nonstandard Analysis, Kluwer, Dordrecht 1994. [25] B. O'Neill, Semi-Riemannian Manifolds, Academic Press, 1983. [26] L. Schwartz, Sur L'impossibilite de la Multiplication des Distributions, C.R. Acad.Sci. Paris, 239 (1954), 847-848. [27] J. A. Vickers, J. Wilson, Invariance of the distributional curvature of the cone under smooth dieomorphisms, Class. Quantum Grav. 16 (1999), 579-588. [28] J. A. Vickers, Nonlinear generalised functions in general relativity, in: M. Grosser, G. Hormann, M. Kunzinger, M. Oberguggenberger (Eds.), Nonlinear Theory of Generalized Functions, 275-290, Chapman & Hall/CRC, Boca Raton 1999. Electronic Mail: M.G.: michael@mat.univie.ac.at M.K.: Michael.Kunzinger@univie.ac.at R.S.: Roland.Steinbauer@univie.ac.at J.V.: jav@maths.soton.ac.uk 24